# §18.19 Hahn Class: Definitions

## Hahn, Krawtchouk, Meixner, and Charlier

Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and normalization (§§18.2(i), 18.2(iii)) for the Hahn polynomials $\mathop{Q_{n}\/}\nolimits\!\left(x;\alpha,\beta,N\right)$, Krawtchouk polynomials $\mathop{K_{n}\/}\nolimits\!\left(x;p,N\right)$, Meixner polynomials $\mathop{M_{n}\/}\nolimits\!\left(x;\beta,c\right)$, and Charlier polynomials $\mathop{C_{n}\/}\nolimits\!\left(x;a\right)$.

## Continuous Hahn

These polynomials are orthogonal on $(-\infty,\infty)$, and with $\Re{a}>0$, $\Re{b}>0$ are defined as follows.

 18.19.1 $p_{n}(x)=\mathop{p_{n}\/}\nolimits\!\left(x;a,b,\overline{a},\overline{b}% \right),$
 18.19.2 $w(z;a,b,\overline{a},\overline{b})=\mathop{\Gamma\/}\nolimits\!\left(a+iz% \right)\mathop{\Gamma\/}\nolimits\!\left(b+iz\right)\mathop{\Gamma\/}\nolimits% \!\left(\overline{a}-iz\right)\mathop{\Gamma\/}\nolimits\!\left(\overline{b}-% iz\right),$
 18.19.3 $w(x)=w(x;a,b,\overline{a},\overline{b})=|\mathop{\Gamma\/}\nolimits\!\left(a+% \mathrm{i}x\right)\mathop{\Gamma\/}\nolimits\!\left(b+\mathrm{i}x\right)|^{2},$
 18.19.4 $h_{n}=\frac{2\pi\mathop{\Gamma\/}\nolimits\!\left(n+a+\overline{a}\right)% \mathop{\Gamma\/}\nolimits\!\left(n+b+\overline{b}\right)|\mathop{\Gamma\/}% \nolimits\!\left(n+a+\overline{b}\right)|^{2}}{\left(2n+2\Re{(a+b)}-1\right)% \mathop{\Gamma\/}\nolimits\!\left(n+2\Re{(a+b)}-1\right)n!},$
 18.19.5 $k_{n}=\frac{{\left(n+2\Re{(a+b)}-1\right)_{n}}}{n!}.$

## Meixner–Pollaczek

These polynomials are orthogonal on $(-\infty,\infty)$, and are defined as follows.

 18.19.6 $p_{n}(x)=\mathop{P^{(\lambda)}_{n}\/}\nolimits\!\left(x;\phi\right),$
 18.19.7 $w^{(\lambda)}(z;\phi)=\mathop{\Gamma\/}\nolimits\!\left(\lambda+iz\right)% \mathop{\Gamma\/}\nolimits\!\left(\lambda-iz\right)e^{(2\phi-\pi)z},$
 18.19.8 $w(x)=w^{(\lambda)}(x;\phi)=\left|\mathop{\Gamma\/}\nolimits\!\left(\lambda+% \mathrm{i}x\right)\right|^{2}e^{(2\phi-\pi)x},$ $\lambda>0$, $0<\phi<\pi$,
 18.19.9 $\displaystyle h_{n}$ $\displaystyle=\frac{2\pi\mathop{\Gamma\/}\nolimits\!\left(n+2\lambda\right)}{(% 2\mathop{\sin\/}\nolimits\phi)^{2\lambda}n!},$ $\displaystyle k_{n}$ $\displaystyle=\frac{(2\mathop{\sin\/}\nolimits\phi)^{n}}{n!}.$